Gradient information of equilibrium quantities is obtained using an adjoint method [1]}, [2]} which requires solving a modified equilibrium problem. For the objectives presented in this work – the rotational transform (§REF ), the magnetic well (§REF ), and near-axis quasisymmetry (§REF ) – the adjoint equilibrium problem requires the addition of a perturbation to the toroidal current profile, pressure profile, and the addition of an anisotropic pressure tensor of a specific form. For other figures of merit, the adjoint equilibrium problem requires the addition of a bulk force of a different form, such as an anisotropic pressure that cannot be handled by the variational principle employed by the ANIMEC code [3]}. In this case, rather than approximating the linearized adjoint problem by adding a perturbation to the nonlinear equilibrium solution, a linearized equilibrium solution can be computed. We have demonstrated this technique for computing the shape gradient of the magnetic well for axisymmetric equilibria [4]}. In generalizing this approach to 3D equilibria, there are additional challenges that arise. As regular singular points occur at every surface where the rotational transform resonates with a mode included in the spectrum for the solution vector, additional care must be taken in regularizing the equations. To avoid these difficulties associated with 3D MHD equilibria, the adjoint equations for a vacuum or force-free equilibrium model could instead be considered.
To conclude, we anticipate many extensions of this work, such as applying the same adjoint principle for free-boundary optimization.
